In short, it's a challenge where you stick with one sim family, not cheating, and marrying townies, maids or whoever, until you reach ten generations... Needless to say, it takes a looong time.
The official rules are here:
The Legacy Challenge
Most people though prefer to play without the points system, or some of the rules... My legacy for example, I don't play it properly, with having the eldest male as a heir and keeping them in the legacy founders house. I'm just playing through ten generations, making them as interesting as possible, and not cheating on the way.